Introduction to Probability and Stochastic Processes

This is a first course in probability and stochastic processes. It introduces the basic concepts of sample space, probability, random variable, independence and conditional probability, along with a variety of common useful distributions. It begins with the intuitive idea of probability based on the frequency of occurrence of an event, then presents the formal mathematical definition of probability, and, eventually arcs back to the intuitive idea with the so-called Law of Large Numbers which shows that, indeed, frequency converges to probability in the appropriate sense.

Under resources you find a link to a website called Random which contains a lot of material in probability that can take you as far as you wish. It also offers applets to run simple experiments. You may use it to complement this material.

A broad strokes introduction with examples of the main topics covered in the course is given in the following video.

Here is a more detailed list of the topics covered:

• Counting Outcomes of Experiments and Sampling
• Sample Spaces, Events, and Probability
• Inclusion-Exclusion Principle
• Monotone Sequences of Events
• Conditional Probability
• Bayes' Formula
• Independence
• The Gambler's Ruin Problem
• Conditional Probability is a probability
• Discrete Random Variables, Expectation, and Variance
• Probability Mass and Cumulative Distribution Functions
• Bernoulli, Binomial, and Poisson Distributions
• Continuous Random Variables
• Probability Density Functions, Expectation, and Variance
• Uniform, Normal, and Exponential Distributions
• Chebyshev's Inequality and the Law of Large Numbers